Duality for group variety For any abelian variety $A$, there is a dual abelian variety $\hat{A}$ which parametrizes degree zero line bundles.
Is it possible to expect similar duality for group varieties (suppose over $\mathbb{C}$ for simplicity)? I noticed that there is a notion of Cartier duality, but I feel this is not what I am looking for. 
Besides, what is wrong if one try to look at the moduli space of degree zero line bundles on group variety just as the case in abelian variety?
 A: The "degree" of a line bundle is not defined in general, it is better to study $\textrm{Pic}^0$, the connected component of the indentity of the Picard scheme. For abelian varieties this is exactly the dual abelian variety.
As noted in the comments, the Picard group of a linear algebraic group $G$ is finite, so $\textrm{Pic}^0(G)=0$ and hence it is not very interesting. By Chevalley's theorem, we may write any algebraic group $H$ as 
$$0 \to G \to H \to A \to 0,$$
where $G$ is linear algebraic and $A$ is abelian. I claim that $\textrm{Pic}^0(H) = \textrm{Pic}^0(A) = \widehat{A}$, so one can reduce to the case of abelian varieties.
Here is a sketch of a proof. By [1, Prop. 6.10], we have an exact sequence
$$0 \to \textrm{Pic}(A) \to \textrm{Pic}(H) \to \textrm{Pic}(G).$$
As already noted, $\textrm{Pic}(G)$ is finite. Hence when we restrict this exact sequence, to the identity component, we obtain $\textrm{Pic}^0(H) = \textrm{Pic}^0(A)$, as required.
[1] Sansuc - Groupe de Brauer et arithmétique des groupes algébriques linéaires.
A: For linear algebraic groups $G$, the picard group is the group of characters on $G$  which is finitely generated (and will be trivial for semi-simple groups).
So one does not talk of moduli space of degree zero line bundles for them.
If you are talking of group varieties that are neither affine nor projective then Chevalley's theorem states that they have unique maximal normal closed linear subgroups making the quotient an abelian variety. So it will reduce to studying on abelian  varieties.
